Signal Processing for Magnetoencephalography

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Signal Processing for

Magnetoencephalography

Rupert Benjamin Clarke

Submitted for the degree of Doctor of Philosophy

University of York Department of Electronics

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Abstract

Magnetoencephalography (MEG) is a non-invasive technology for imaging human brain function. Contemporary methods of analysing MEG data include dipole fitting, minimum norm estimation (MNE) and beamforming. These are concerned with localising brain activity, but in isolation they do not provide concrete evidence of interaction among brain regions. Since cognitive neuroscience demands answers to this type of question, a novel signal processing framework has been developed consisting of three stages. The first stage uses conventional MNE to separate a small number of underlying source signals from a large data set. The second stage is a novel time-frequency analysis consisting of a recursive filter bank. Finally, the filtered outputs from different brain regions are compared using a unique partial cross-correlation analysis that accounts for propagation time. The output from this final stage could be used to construct conditional independence graphs depicting the internal networks of the brain.

In the second processing stage, a complementary pair of high- and low-pass filters is iteratively applied to a discrete time series. The low-pass output is critically sampled at each stage, which both removes redundant information and effectively scales the filter coefficients in time. The approach is similar to the Fast Wavelet Transform (FWT), but features a more sophisticated resampling step. This, in combination with the filter design procedure leads to a finer frequency resolution than the FWT.

The subsequent correlation analysis is unusual in that a latency estimation procedure is included to establish the probable transmission delays between regions of interest. This test statistic does not follow the same distribution as a conventional correlation measures, so an empirical model has been developed to facilitate hypothesis testing.

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5.4.1 Time-domain View . . . 91 5.4.2 Frequency-domain View . . . 98 5.5 Summary . . . 116 6 Statistical Framework 119 6.1 Theory . . . 119 6.1.1 Correlation . . . 120 6.1.2 Partial Correlation . . . 121 6.1.3 Cross Correlation . . . 122 6.2 A Test Statistic . . . 123 6.2.1 Latency Estimate . . . 125 6.2.2 Null Model . . . 125

6.3 Higher order statistics . . . 129

6.4 Application . . . 135 6.4.1 Introduction . . . 135 6.4.2 Methods . . . 136 6.4.3 Results . . . 146 6.4.4 Discussion . . . 146 6.5 Summary . . . 150

7 Summary and Further Work 152 7.1 Summary . . . 152

7.1.1 Aims . . . 152

7.1.2 MEG in the context of Neuroimaging . . . 152

7.1.3 MEG Analysis . . . 153

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Contents 6

7.1.5 Connectivity Analysis . . . 154

7.2 Further Work . . . 155

7.2.1 Source Signal Estimation . . . 155

7.2.2 Time-frequency Decomposition . . . 155

7.2.3 Connectivity Analysis . . . 156

A Spherical Head Model 157

Abbreviations 162

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List of Figures

2.1 Gross anatomy of the brain . . . 15

2.2 Lateral view of cerebrum . . . 16

2.3 General morphology of a neuron . . . 19

2.4 The first MEG measurement, made in 1971 . . . 23

2.5 MEG System at York Neuroimaging Centre . . . 24

2.6 Experimental current dipole . . . 36

3.1 Data flow diagram of MEG signal processing framework . . . 50

3.2 Conditional independence graph . . . 51

4.1 Representative diagram of an L-curve . . . 60

4.2 MNE solutions from simulated dipoles . . . 69

5.1 Time and frequency domain representations of a MEG signal . . . 73

5.2 Effect of windowing on DFT . . . 78

5.3 Spectral leakage in DFT due to rectangular windowing . . . 79

5.4 Sidebands in Fourier spectrum introduced by rectangular window . . . 80

5.5 Spectrogram of linear chirp . . . 82

5.6 Two common mother wavelets . . . 85

5.7 Multiresolution Analysis . . . 89

5.8 Hann window . . . 93

5.9 Sequence of raised cosine wavelets . . . 93

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List of Figures 8

5.11 Fourier transform of raised cosine wavelets . . . 98

5.12 Transfer function of raised cosine wavelet . . . 99

5.13 Asymmetrical slopes of adjacent wavelet scales . . . 99

5.14 Impulse responses of ideal and windowed filters . . . 103

5.15 Illustration of filter design and action upon MEG signal . . . 107

5.16 Impulse response of filter bank . . . 112

5.17 Examples of filter bank output . . . 113

5.18 Reconstruction of input signal . . . 114

5.19 Short-time Fourier transform . . . 115

5.20 Change in power with respect to baselines . . . 116

6.1 Bootstrapped test statistic with noise input . . . 128

6.2 Trends in null distribution of test statistic vs. sample size . . . 130

6.3 Trends in null distribution of test statistic vs. sequence length . . . 131

6.4 Cross correlation with simulated data . . . 133

6.5 Removal of confounding influence using PCA . . . 134

6.6 Approximate locations of auditory and speech areas on inflated brain . . . . 136

6.7 Location of chosen regions of interest . . . 138

6.8 Minimum norm estimates from averaged data . . . 139

6.9 Epoch average of gamma activity in regions of interest . . . 141

6.10 Left hemisphere amplitude changes (A & B) . . . 142

6.11 Left hemisphere amplitude changes (C1& C2) . . . 143

6.12 Right hemisphere amplitude changes (D & E) . . . 144

6.13 Right hemisphere amplitude changes (F) . . . 145

6.14 Interaction model during auditory stimulation (35.3 – 44.1 Hz) . . . 147

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Acknowledgements

I owe a considerable debt of gratitude to the many people without whose support this thesis could not have been produced. It would be impossible for me to name everyone who has contributed in one way or another, but the following deserve special mention:

Gary Green David Halliday Aziz Asghar P´adraig Kitterick Andre Gouws Will Woods Mark Hymers Ash Jansari

and all the staff at YNiC.

Financial Assistance

This PhD was generously funded by the York Neuroimaging Centre (YNiC) in partnership with the University of York. I am grateful for their support.

Declaration

The work presented in this thesis is entirely that of the author except where otherwise indicated. No part of this document has been submitted for previous examination.

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1 Introduction

Magnetoencephalography (MEG) is a non-invasive functional neuroimaging modality that is closely related to the long-established practice of electroencephalography (EEG). When neurons are activated in living organisms, ionic currents flow, producing extremely low-intensity magnetic fields. This phenomenon is called neuroelectromagnetism. The technol-ogy is available to record these fields externally and produce graphical representations of them. The primary subject of measurement is the human brain.

There are two broad areas of application for MEG: clinical assessment and cognitive neuroscience [1]. In the former application, epilepsy patients with intractable seizures that do not respond to drug therapy are examined using MEG prior to neurosurgery [2]. Here, the clinician is interested in locating the focus of epilepsy. In this respect, MEG offers heightened accuracy over EEG, albeit at greater expense. In the second application, cognitive psychologists make use of MEG to study and characterise dynamic brain activity in both healthy and diseased states.

Commercial MEG systems have been available for over 20 years, and compared with other functional neuroimaging methods, MEG is minimally invasive while achieving comparable spatial resolution and significantly superior time resolution [1, 3]. In view of this, it may seem surprising that MEG systems are not more commonplace than is actually so. One explanation is that the field of MEG analysis is impoverished of suitable techniques for answering the questions posed by neuroscience. In particular, while the problem of locating neuromagnetic sources within the brain has been addressed in many ways, relatively little has been done to assist the neuroscientist with making inferences from that information. One might wish to infer, for example, that certain stimuli cause increased

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activity in a particular brain region. In the search for potential areas of novel contribution in MEG analysis, the present author concluded that a combination of selected digital signal processing algorithms and statistical analysis might offer additional insight in this respect. This idea formed the basis for the project described herein.

In this thesis, an engineered approach to connectivity analysis is proposed, which consists of using an existing source estimation process to extract signal estimates from a small number of regions of interest in the cerebral cortex. This is to be followed by a time-frequency analysis that subdivides neurophysiological frequency bands. Finally, the spectrotemporal representation shall be subjected to statistical analysis to determine the associations between regions of interest. In the course of developing the proposed framework, a novel time-frequency decomposition was also developed consisting of a filter-bank made up of cascaded low-pass finite impulse response (FIR) filters. It is possible to tune the characteristics of the filter-bank, but a particular tuning was suggested which subdivides traditional neurophysiological frequency bands by a factor of approximately 3. The final development of the approach was a connectivity analysis that was based upon partial correlation methods.

The earlier chapters of this thesis will provide a fundamental background in neuroscience, MEG recording and contemporary MEG analysis in order to contextualise the subsequent content. A discussion of some concepts for novel approaches to MEG analysis is then presented. Finally, the full implementation of one particular concept is described in detail over several chapters, with examples of its application to typical experimental data. The tools developed are designed to contribute to furthering understanding of human brain function.

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1.1 Chapter Overview 12

1.1 Chapter Overview

The remaining chapters of the thesis are described below.

Background In this chapter, the underpinning knowledge required to understand the

techniques discussed in the thesis is presented. Established forms of MEG analysis are also considered.

Concepts This chapter describes some research topics that were considered, forming

a philosophical argument for devloping a new MEG analysis framework, which is presented last.

Minimum Norm Estimation This chapter comprises a detailed study on the theory and

practice of minimum norm estimation, which is an important component of the proposed framework. Some examples of its use are included, based on simulated data.

Time-frequency analysis An argument for using time-frequency representations in the

interpretation of MEG data is presented in this chapter, followed by a review of existing techniques. A novel approach to time frequency analysis is then developed, with examples of its application to MEG data. This constitutes the second stage of the framework.

Statistical Framework This chapter charts the development of a suite of tools for

analysing MEG data preprocessed using the techniques described in the previous two chapters. The integration of all of the techniques making up the framework is discussed, with a preliminary example of its application to MEG data.

Summary and Further Work A summary of the work presented throughout the thesis is

given. Additional work that could be carried out to continue and enhance the concepts that have been developed is then suggested.

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2 Background

Magnetoencephalography is one of a number of functional neuroimaging modalities, which otherwise include Electroencephalography (EEG), functional Magnetic Resonance Imaging (fMRI) and Positron Emission Tomography (PET). Functional neuroimaging is concerned with the detection of time-dependant activity within the signaling systems of the nervous system. Such measurements are intended to elucidate the role of the brain in cognition and behaviour, motor function, endocrinology and autonomic functions such as breathing, circulation and digestion. This chapter provides the appropriate theoretical and historical background to the work described later in the thesis.

2.1 The Human Brain

For the purposes of understanding the requirements of an MEG analysis method, it is useful to first consider the anatomy of the brain and its general function. The first part of this section studies the gross anatomy of the brain. Proceeding this is a section that examines the microscopic elements that make up the cerebral cortex, and how these are thought to contribute to MEG signals.

2.1.1 Macroscopic View

The brain is categorised as part of the central nervous system (CNS), which also includes the spinal cord. Sensory information from receptors around the body arrives at the spinal cord via afferent nerves. Motor function is brought about by efferent nerves that join the spinal cord to remote effectors. Together, these nerves make up the peripheral nervous

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2.1 The Human Brain 14

system. From the point where the peripheral nerves join the spinal cord, further connections ascend to the brain stem. Here, they mostly decussate, i.e. they cross from left to right and vice versa. Consequently, sensory and motor function from one side of the body is associated with the brain hemisphere on the opposite side. Descending (motor) nerves pass through the internal capsule within the midbrain, having originated in the motor area of the cortex, the precentral gyrus. Ascending nerves terminate in the thalamus, where further connections are made to the postcentral gyrus, which processes somatosensory information — sensations of pressure, temperature, pain etc [4].

Brain

From an embryological point of view, the brain develops in three sections: the forebrain, midbrainand hindbrain [4]. The forebrain splits into two halves that curve over to form the cerebral hemispheres. These envelope the deeper structures of the forebrain (the limbic system). The midbrain forms part of the brainstem, joining the basal ganglia of the cerebral hemispheres to the hindbrain. The hindbrain forms the medulla, pons and cerebellum. Figure 2.1 shows the major structures in the brain.

The cerebral hemispheres consist of an outer layer of grey matter with many folds. This is known as the cerebral cortex. Grey matter is so-called because it consists mostly of cell bodies, which appear grey in a preserved brain [4]. Underneath is a core of white matter, made up of myelinated nerve fibres, which are bundles of axons. The myelin sheaths are electrically insulating, and serve to isolate nerves from each other [4]. Some nerves, known as association fibres, connect different parts of the cortex within the same hemisphere. Projection fibres make connections between the cerebral cortex and either subcortical structures, the brainstem or spinal cord. Lastly, commissural fibres connect equivalent regions within the two hemispheres [4]. A large body of commissural fibres exists, referred to as the corpus callosum.

The inferior (deeper) parts of the cerebral hemispheres which surround the brainstem, are known together as the limbic system. This includes the cingulate gyrus, which is involved

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2.1 The Human Brain 15 corpus callosum cingulate gyrus fornix cerebral cortex cerebellum pons medulla

(a) Medial sagittal section

thalamus

hippocampus amygdala

(b) Medial sagittal section with selected inferior lateral structures overlayed

Figure 2.1: Gross anatomy of the brain

with autonomic functions such as heart rate, as well as many other functions including attention [5]. Also included are the hippocampus and parahypocampal gyrus. These are involved with memory [5]. The hypothalamus connects the brain with the endocrine system, thus affecting the production of hormones. The thalamus provides an interface between the cortex and other parts of the CNS. The fornix is composed of white matter, and connects the hippocampus to both the mamillary bodies and part of the cortex. Finally, the amygdala is located in the temporal region and is implicated in emotional function. In general, the limbic system as a whole is associated with emotion, learning and memory [5].

The cerebellum is a large neuronal structure immediately below the cerebral hemispheres to the rear of the brainstem. Thick fibre tracts attach it to the pons. Its function is predominantly related to movement and proprioception (the sense of movement and location) [5].

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Cerebral Cortex

In MEG studies, the cerebral cortex is particularly important because MEG is primarily sensitive to superficial sources due to their proximity to the sensors. It consists of considerable numbers of neurons of two major types: pyramidal cells and stellate cells [4]. The densities of these cells varies throughout the cortex, contributing to functional specialisation. Pyramidal cells are elongate structures possessing an apical dendrite. This is a long dendrite from which several shorter branches emerge, receiving very many synaptic connections from other neurons (see §2.1.2). The apical dendrite usually extends towards the cortical surface. Pyramidal cells vary in height from about 10 µm up to 100 µm for some neurons in the motor cortex. The axons of pyramidal cells connect to other brain regions. In comparison, stellate cells are more rounded with no apical dendrite. They mainly serve to make short connections between neurons, each with relatively few synapses.

The many folds that appear in the cerebral cortex are called sulci [4]. The convex regions between them are gyri. Although the cortex has very many small sulci, there are a few much larger ones, sometimes called fissures, which divide the cortex into distinct lobes. The four lobes, shown in figure 2.2, are the frontal, parietal, occipital and temporal lobes. As well as being physically separated, these lobes are functionally distinct.

Figure 2.2: Lateral view of cerebrum

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2.1.2 Microscopic View

The microanatomy of the CNS generally concerns two types of cell from which it is composed. Those are neurons and glial cells (neuroglia) [4, 5]. Neurons are the decision-making units in the CNS. Neuroglia do not directly take part in information transmission or processing. However, they are vital to the operation of the CNS. This section describes the morphology of neurons and neuroglia, as well as the mechanism of communication between neurons.

Morphology

Neurons feature various narrow projections (cell processes) leading away from the cell body (soma) [4, 5]. Axons are the processes that carry signals to other neurons. They are frequently sheathed in myelin, a fatty substance that is an electrical insulator. They terminate at a number of synapses, which are the connections to other neurons. Those connections are made onto dendrites, further neuronal processes that receive signals. Dendrites have many branches, the most extreme of which have the highest concentration of protruding spines that are the postsynaptic processes (synaptic receptacles). Unlike axons, they are never myelinated. Within the soma is a nucleus and various organelles, which take part in the development and metabolism of the cell. The cell, including its processes, is otherwise filled with a continuous body of cytoplasm. Figure 2.3 depicts the structure of a neuron.

The neurons fall into four categories depending on their form [4, 5]. Unipolar neurons have a single process extending from their soma, which branches to form both dendritic and axonal terminals. Bipolar neurons have a separate axon and dendrite extending in different directions. Pseudounipolar neurons are similar to unipolar neurons in appearance, but are in fact bipolar neurons whose dendrite and axon are fused together. Finally, multipolar neurons are those with a considerable branching dendrite structure (the dendritic field), which immediately surrounds the soma. They also possess an axon that extends away from the soma and can potentially synapse with distant neurons. This axon branches rarely, if at

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all, over most of its length. Multipolar neurons are the type usually occurring in the brain. Although neurons form the signaling network of the CNS, there are several times as many neuroglia present. These fall into three categories [4, 5]. Astrocytes are approximately rounded cells with numerous branches extending from the soma. They surround neurons, serving to provide nutrients to them. Notably, they form the blood-brain barrier. This transports ions between the blood and extracellular fluid of the brain, while rejecting blood-borne agents that may disrupt the function of the neurons. Oligodendrocytes are myelin-producing cells, with each one myelinating several axons in spiraling concentric laminations. Microglia are phagocytes that remove the debris of damaged tissue.

In addition to the myelin produced by oligodendrocytes, another type of myelination is performed by Schwann cells [4, 5]. Each Schwann cell only myelinates part of a single axon. Whether myelinated by Schwann cells or oligodendrocytes, the myelinated regions are periodically interrupted, exposing part of the axonal membrane. This exposed part is called a node of Ranvier, and serves to enhance the transmission of action potentials as described in the next section.

Signaling

Signaling between neurons takes place by way of the synapses. A single synapse consists of three parts. This first component is the presynaptic terminal, which is one of the synaptic boutons at the extremities of an axon. The second is the synaptic cleft, a gap of approximately 20 nm that occurs between the presynaptic terminal and the remaining component of the synapse, the postsynaptic process. As previously stated, this is usually a dendritic spine occurring on another neuron. If this is the case then it is called an axodendritic synapse. Synapses can also connect directly onto the soma of the receiving neuron, in which case they are classed as axosomatic. Other types of synapse do exist (e.g. dendrodendritic), but in far smaller numbers [4, 5].

The cytoplasm within a neuron is at a lower electrical potential than the extracellular fluid [5]. If the extracellular fluid is defined as being at 0V, the voltage of the cytoplasm

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2.1 The Human Brain 19 Schwann cells (myelination) axon terminals dendrites axon hillock soma axon nodes of Ranvier

Figure 2.3: General morphology of a neuron

is negative. This negative voltage is known as the membrane potential as it occurs across the cell membrane. In the neuron’s resting state, the membrane potential is maintained at about -70 mV by active ion pumps in the neuronal membrane [5]. These pump sodium (Na+) ions out of the cell and potassium (K+) ions into it, in a 3:2 ratio. Thus the intracellular concentration of K+increases while that of Na+decreases. The 3:2 imbalance causes the neuron to become negatively charged. An equilibrium is maintained because the ion pumps act against diffusion and passive electrical currents, which cause a limited opposing transport of ions through non-gated ion channels. A voltage is developed across the membrane due to the diffusion of K+down the concentration gradient. The cell is said to be polarised.

Changes in the postsynaptic membrane potential occur when chemical neurotransmitters are released from the presynaptic terminal into the synaptic cleft [5]. Such releases are caused by the firing of the presynaptic neuron. The subsequent change in membrane potential is brought about by the action of gated ion channels. These respond to the

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neurotransmitter by allowing movement of ions across the postsynaptic membrane. This action is “graded” — the ionic current (and its resultant effect on membrane potential) varies in magnitude according to the neurotransmitter concentration [5]. Also, the direction of the current varies between synapses, depending on whether they are excitatory or inhibitory. Excitatory synapses raise the membrane potential (decreasing its magnitude) while inhibitory synapses lower it. It should be noted that the membrane potential is not constant throughout the cell due to the resistivity of the cytoplasm. As such, changes in membrane potential will diminish with distance from the ionic current source that caused them.

In the axon hillock (where the axon emerges from the soma) and the axon itself, voltage-dependent ion channels exist in the cell membrane [5]. If the membrane potential exceeds a particular threshold, these channels allow Na+ ions to enter the cell. The ions will do so due to the potential difference and the concentration gradient created by the ion pumps. The influx of Na+ ions further depolarises the neuron, increasing the membrane potential and opening yet more Na+channels. The resulting avalanche in depolarising ionic currents causes the membrane potential to spike, generating an action potential that continues down the axon [5]. Once the cell is depolarised, K+channels open, repolarising the neuron. In fact, the repolarisation overshoots briefly, during which time a second spike is less likely to occur (the membrane potential is further than usual from the threshold). Also, the Na+ channels are briefly inactivated. Both factors limit the rate at which action potentials can be generated.

The magnitude of the action potential decays exponentially along the axon as the volume currents diverge. This limits both the distance that the signal can travel, and the speed. At successive gated ion channels, the rising edge of the action potential takes longer to exceed the threshold because of the attenuation. However, in myelinated axons, this effect is mitigated by the presence of the myelin [5]. The fatty coating insulates sections of the axon such that the voltage dropped along the axon is reduced, through the elimination of volume currents flowing out of the cell. The nodes of Ranvier provide points at which the action

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2.2 Magnetoencephalography 21

potential is regenerated by voltage-dependent, gated ion channels [5]. The transmission speed is increased because the signal effectively jumps from one node to the next via the low-loss pathway.

As well as synapses mediated by neurotransmitters, a further type of synapse exists that uses electrical instead of chemical transmission [5]. In these synapses, the presynaptic and postsynaptic membranes are in very close contact. Electrical transmission is facilitated by gap junctions [5]. These are groups of protein molecules situated in the pre- and post-synaptic membranes, which form pores that result in the continuity of cytoplasm from one neuron to the next. Electrical synapses permit much quicker communication between neurons than chemically mediated synapses. Most are also bidirectional.

MEG signals are thought to be primarily due to dendritic currents [6, 7]. Axonal current sources consist of a depolarisation front that represents the commencement of an action potential and travels down the length of the axon, followed closely by a repolarisation front. The separation of the two depends on the duration of the action potential, but is consistently very small. This can be modelled by two oppositely-polarised current dipoles (see §2.3), forming a quadrupole. Quadrupolar sources produce higher order spatial fields that drop off rapidly with distance, thus contributing very little to the signal measured by MEG [7].

2.2 Magnetoencephalography

This section is intended as a primer on MEG. It includes a brief history of neuroelectromag-netic measurement, followed by a description of the instrumentation used in MEG and its use. Rival modalities are also considered and the place of MEG in functional neuroimaging is then established.

2.2.1 History

The field of medicine has been aware of the role of electricity in nerve fuction since the 1700s, owing to pioneers such as Luigi Galvani, whose stimulation of frog muscles in 1781 firmly established the connection between electrical stimulation and muscle

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contraction [8]. The subsequent invention of the galvanometer the following century facilitated the measurement of endogenous currents in frog muscles, firstly by Carlo Matteucci in 1838. Then Emil du Bois-Reymond characterised the nerve impulse or action potentialin 1841 [9].

In the second half of the 19th century, theories on the structure and organisation of the central nervous system began to emerge. John Hughlings-Jackson proposed that brain regions might have different functions after noting the ordered progression of muscle contractions during epileptic seizures [5]. Marc Dax, Pierre Paul Broca and Carl Wernicke all used autopsy to identify regions of the cerebral cortex that, when damaged by stroke, impaired language ability [10, 5]. These discoveries lent credence to the notion of functional specialisation in the brain.

The development of histological staining of tissue samples by Camillo Golgi in 1873 allowed Santiago Ram´on y Cajal to produce highly-renowned illustrations of neuronal structures [9]. Using similar techniques, Korbinian Brodmann categorised 52 distinct regions of the cerebral cortex based on differences in microanatomy in 1909 [5].

Meanwhile, Richard Caton had conducted invasive intra-cranial EEG on animal cortices, having begun investigations in 1875. Many other investigators followed suit, but not until 1924 did Hans Berger carry out the initial extra-cranial EEG measurements on human beings. Berger succeeded in identifying alpha and beta oscillations with his technique, which was brought to the attention of the medical fraternity when these results were replicated by Adrian and Matthews in 1934 [9].

The first biomagnetic measurements were magnetocardiographic (MCG) measurements carried out by Baule and McFee in the 1960s [11]. For this, they used two similar coils on a ferrite core, each with a considerable number of turns. These were wired in series with opposite polarity, to form a type of gradiometer that would detect the nearby biomagnetic field but not the distant environmental noise fields. The measurements were made outdoors, as far as possible from noise sources. This approach was successful, but produced noisy results. Not long afterwards, David Cohen began making similar measurements, this

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time using a low-noise amplifier and a magnetically shielded environment, with improved results [12]. Using this set-up, Cohen also attempted the first MEG measurements in 1968, but the signal-to-noise ratio was very poor. Then in 1969, a device known as the SQUID (Superconducting QUantum Interference Device, §2.2.2), was invented by Jim Zimmerman. The SQUID is a very low noise detector, and when Cohen learnt of this development, a collaboration was organised with Zimmerman on the biomagnetism research [1]. They succeeded in producing a very clean MCG trace using one of Zimmerman’s experimental devices. By 1971, commercially manufactured SQUIDs had become available, which Cohen employed to make the first MEG recording [13] (Fig. 2.4). The outstanding results prompted widespread interest in magnetoencephalography.

Figure 2.4: The first MEG measurement, made in 1971 (taken from [12])

2.2.2 Instrumentation and Operation

Modern MEG equipment measures neuromagnetic fields using several hundred magne-tometers, closely arranged around the subject’s scalp. The centrepiece of a whole-head MEG system is usually a large housing incorporating a helmet-shaped cavity. The subject’s head rests inside the cavity while recording takes place. Internal magnetometers are located immediately around the helmet so as to minimise their distance from the neuromagnetic source, yielding the strongest possible signal. Figure 2.5 shows the MEG system used to capture the results presented in the later chapters.

Neuromagnetic fields arise mainly from postsynaptic currents that flow within the dendrites of neurons, as discussed in §2.1.2. The magnetic field of individual action potentials in the brain is not measurable using MEG. The smallest extra-cranial magnetic

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This MEG system (4-D Neuroimaging Magnes 3600WH) contains 248 primary measurement coils, with 28 additional sensors for measuring environmental noise fields. The angle of the Dewar can be adjusted to accommodate measurements with the subject in seated or supine positions. In the photograph, the Dewar is in an intermediate position that reduces the boil-off of liquid helium when not in use. Photograph: R.B. Clarke

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fields that can be detected are the combined effect of more than 104 near-simultaneous neuron activations [14]. This signal is representative of neuronal activation in general because the highly-connected neurons are inclined to become active together in localised groups. Even so, the fields are only on the order of tens of femtoTesla [14], which is so minute that cryogenically-cooled SQUID magnetometers are required to achieve adequate signal-to-noise ratios. The majority of the MEG housing encloses a Dewar containing the liquid helium cryogen that supports the operation of these highly sensitive devices.

A SQUID is a Superconducting Quantum Interference Device, a specialised magnetome-ter having much lower output noise than conventional designs. MEG uses d.c. SQUIDs, which consist of a superconducting ring with two Josephson junctions [3, 1]. This is simply a ring made from a particular alloy, interrupted by miniscule gaps that are the Josephson junctions. Electrons can cross the junctions due to quantum tunnelling effects, with a phase shift developing across the junction. The superconducting property of the alloy is only achieved at a very low temperature. A cryogenic environment at 4.2K surrounds all components of the sensors. Neuromagnetic fields induce current in a pick-up coil, which is connected across a complementary signal coil, forming a flux transformer. The signal coil is then inductively coupled to the SQUID, which is connected across the input of a low noise amplifier. The SQUID is biased with a direct current. Then, due to quantum phase effects, the voltage across the SQUID becomes a periodic function whose amplitude depends on the magnetic flux coupled to the SQUID. It is this varying amplitude (not the periodic signal) that becomes the MEG signal.

The neuromagnetic fields being measured are many orders of magnitude smaller than typical ambient magnetic fields, such as the geomagnetic field of the planet Earth (which is about 50µT). MEG equipment must be sensitive enough to measure the relevant neurological fields, but insensitive to much larger noise signals. Problematic noise signals can originate from several sources. These include the ambient magnetic fields of the physical environment, the intrinsic noise of the sensors, and the input noise of the amplifiers receiving the sensor signals. MEG systems employ specific design features to combat each

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of these potential sources of noise. Firstly, environmental noise fields are greatly attenuated by housing the complete measuring instrument in a magnetically-shielded room. Secondly, SQUID magnetometers are utilised for their superior noise performance. Lastly, the output from the SQUIDs is amplified using low-noise instrumentation amplifiers. These features can lead to spectral noise densities of less than 5fT/√Hz [15]. Many systems implement analogue and digital noise reduction to further improve the signal-to-noise ratio.

During MEG investigations, the output from every magnetometer (or occasionally a subset of them) is recorded over time. Stimuli such as sounds (auditory stimuli) or images (visual stimuli) are often presented to the subject during the recording to investigate neurological responses. Spontaneous brain activity can also be measured. Recent MEG systems digitise the signals for storage and manipulation. After the recording, the data may be analysed using various processes to establish the location and time course of neuronal activity. The spatial resolution can be better than 3 mm [3, 14]. However, this depends on several factors including the analysis method, number of sensors, the location of the activity and the presence of noise. MEG analysis methods will be addressed in §2.3.

Temporal resolutions in MEG range down to 1 ms. The technology might allow for a greater bandwidth, but there is a practical limitation on the highest frequencies that can be detected in the presence of noise. The power spectrum of most noise signals is white, but the neuromagnetic signals diminish with increasing frequency [16], eventually disappearing below the noise floor. Noise is a constant consideration in experimental design for MEG. Signal averaging over multiple trials is commonly used to improve noise rejection [7]. The data are captured in short time segments, known as epochs. Each epoch captures data from a single stimulus presentation, and often for brief periods before and afterwards. In subsequent offline analysis, the average is taken across sets of similar epochs that were gathered under a particular experimental condition. The time variable is usually referred to the stimulus onset, so signals that are phase-locked to the stimulus are coherent between trials. Averaging attenuates incoherent signals, such as noise, by the square root of the number of trials [7].

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2.2.3 Alternatives

This section briefly considers the main non-invasive functional neuroimaging modalities that complement MEG. These are positron emission tomography (PET), functional mag-netic resonance (fMRI), and electroencephalography (EEG). These modalities are compared with MEG in terms of suitability for particular applications. Some of the practical and safety considerations associated with their use will also be discussed.

PET

A metabolically active compound that has been labelled with a radioactive tracer is administered to the subject intravenously or sometimes by inhalation. The tracer is an isotope with a short half-life that undergoes β+ decay, such as oxygen-15. When a positron is emitted, it travels a very short distance before it collides with a nearby electron. The two are anihilated and a pair of photons are emitted at nearly 180◦ to one another. Some of these photon pairs arrive at a ring of detectors in the scanning equipment. The arivals of photon pairs are correlated in order to reconstruct an image showing how the molecule or its metabolites are distributed in the plane of the detector ring. This gives an indication of localised metabolic rate, and hence the intensity of neural activation [17]. In neuropsychological applications the rate of metabolism is a comparative measurement against a baseline rate, i.e. changes rather than absolute levels are of interest. Some clinical applications do study the absolute levels since abnormal rates of metabolism may indicate pathology [14].

The spatial and temporal resolution of PET depend on the tracer that is employed, but resolutions on the order of a few millimetres and response times of approximately one minute can be achieved. A distinct advantage of this modality is the ability to target a particular neurotransmitter system by labelling an appropriate precursor compound [14]. Because it involves ionising radiation, the use of PET must be strictly limited in order to minimise the radiation dose received by the subject. Radiation exposure is more acceptable in clinical diagnosis where there is foreseeable benefit to the patient. Conversely, PET

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for research purposes is highly restricted. Compared with fMRI, EEG and MEG, PET is regarded as being more invasive due to the requirement to administer radiopharmaceuticals.

fMRI

Functional MRI is a specialised form of magnetic resonance imaging, a modality normally used to provide structural images of the anatomy. In this specialised application of MRI, the technology is adapted to produce tomographic images of changes in haemodynamic responses. These are believed to be the markers of temporal changes in neural activity. MRI in general relies on the phenomenon of nuclear magnetic resonance, which is briefly explained in the following paragraphs. A more detailed discussion of the relevant theory can be found in [18], from which the following information was taken. After the general treatment of MRI, there follows a dedicated paragraph on fMRI.

A subatomic particle has a spin angular momentum associated with it, even if it is stationary. This means it is effectively rotating about its centre of mass. If the particle is charged, the movement of charge results in a circular current flow, which produces a magnetic dipole. A magnetic dipole can be described in terms of the dipole moment, which is a vector having a direction and magnitude. When subjected to a strong uniform magnetostatic field, the dipole moment will precess about the direction of the field, in accordance with Larmor’s Theorem. It will also tend to align with the field1. The rate of precession is proportional to the strength of the field. More importantly, it is also proportional to the ratio between the charge on the particle and its mass, or equivalently, the ratio between the dipole moment and the angular momentum of the particle. This ratio is known as the magnetogyric ratio, and the corresponding frequency is the Larmor frequency. Once a particle is precessing at the Larmor frequency, then if it is further subjected to a second, much smaller, orthogonal magnetic field that rotates about the static field with equal frequency, then the dipole moment will also precess about that field as it rotates.

1

To be entirely accurate, the particle’s dipole will either align with the magnetic field or counter to it. However, in a large group of similar particles, a slight majority will be in alignment with the field, so this is the net effect.

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Thus, the dipole moment is deflected away from the direction of the static field at a rate proportional to the smaller field’s strength and the particle’s magnetogyric ratio. Clearly, the deflection is much slower than the original precession due to the smaller field strength. Other types of particle that are precessing about the static field at a different frequency are not significantly affected. Thus a means of selectively deflecting particles with a particular magnetogyric ratio is provided. In practice, if the strong static field is being produced by a powerful solenoid, the rotating field can be produced by a single perpendicular transmitter coil. Only one coil is required to produce the rotating field by recognising that a sinusoidal field is equal to the sum of two fields rotating in opposite directions, such that one rotates at a frequency which is the negative of the other. Therefore, a radio-frequency (r.f.) sinusoidal signal is used to drive the coil. The field rotating at the negative frequency differs from the Larmor frequency by twice its value, so has no significant effect on dipole precession.

If the rotating field is applied continuously, the dipole continues to be deflected until it is counter-aligned with the static field, before returning to alignment. In this way, it oscillates in and out of alignment with the static field at the frequency determined by the magnetogyric ratio of the particle and the magnitude of the rotating field. The electromagnetic field produced by this particle spin is maximal when the deflection is 90◦. If considerable numbers of similar particles undergo this process, a signal is detectable. In essence, this can be used to measure the abundance of a particular particle within a sample.

By applying an r.f. pulse exactly long enough to deflect the dipole moments through 90◦, the resulting signal from the dipoles continues to be emitted once the pulse is removed. However, the signal does not last indefinitely. The additional energy given to the particle dissipates due to thermal conduction, and the dipole moment returns to alignment with the static field. The signal decays exponentially, with a time constant T1, the longitudinal

relaxation time. A second form of exponential decay occurs because slight inhomogeneities in the static field cause the dipole moments to precess about the static field at slightly different rates. They point in the same direction (perpendicular to both fields), immediately after the 90◦pulse is applied, but as time goes on, they spread out in the plane perpendicular

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to the static field. The resulting decay is described by the transverse relaxation time, T2.

If a second r.f. pulse twice as long as the first is applied after a chosen interval, τ , the dipole moments are deflected through a further 180◦. The relative offsets of the dipole moments due to the transverse spreading are reversed, so they begin to converge as they continue to precess about the static field. After a further duration of τ has elapsed, the dipoles become exactly aligned as they were at the end of the 90◦ pulse. Therefore, the emitted signal reaches a maximum at this time, called the spin echo. It is much easier to measure this signal because it occurs in the absence of any excitation pulses, and at a known time.

In order to make structural images using nuclear magnetic resonance, a gradient is introduced in the static field. The r.f. pulses are tuned to the Larmor frequency of protons located at a particular displacement along the field gradient. The magnetic dipole moments of protons in this narrow slice will be selectively deflected, so the density of protons in that slice can be determined. Three-dimensional images can be resolved by taking numerous different slices and repeating the process with the gradient occurring in the three orthogonal directions.

Functional MRI uses the same technology to produce time-varying images of functional activation in the brain. In a process called the haemodynamic response, oxygenation of blood in the brain responds to changes in local neuron activation by overcompensating for the increase in metabolism that it causes in the region of the neurons. This fact is employed by fMRI in that oxyhaemoglobin is diamagnetic but deoxyhaemoglobin is paramagnetic and possesses a much larger magnetic dipole moment. The local magnetic field is affected so as to reduce the relaxation time of nearby protons. The effect on the spin echo can be detected and used to determine changes in the oxygenation level of the blood in different regions of the brain, and thus the functional activity.

It takes a matter of seconds for the haemodynamic response to reflect an increase in neural activity, which limits the useful temporal resolution of fMRI. Consequently, improvements are generally sought in the spatial resolution, where 3×3×4 mm elements

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are typical [14]. Due to the superior temporal resolution, the technology is more suitable than PET for detecting the subtleties of functional mapping in the brain. The reliance of both on metabolic processes rather than direct electrical activity is a limitation not present in either EEG or MEG. Because strong magnetic fields and r.f. electromagnetic radiation are used in fMRI, electronic and metallic implants are contraindications for undergoing a scan. Otherwise, the procedure is considered harmless because it has no influence at the molecular level.

EEG

In the most common form of EEG, a number of electrodes are applied to the subject’s scalp, where they are used to measure surface electrical potentials at several locations simultaneously. Typically, a cap is placed over the scalp that has apertures at standardised electrode locations. A conductive gel is injected through each hole prior to insertion of a needle electrode, forming a good electrical contact with the scalp. An international standard exists for the location and name of EEG electrodes, called the 10-20 system [8]. This defines 21 electrodes based on the proportion of the distance around the skull at which they are placed. These distances are measured from the nasion point above the nose and between the eyes to the inion point, which is central to the back of the head where there is a bony protrusion at the bottom of the skull. The electrodes are located at the intersections of a grid of lines that are separated by 10% or 20% of the distance around the skull.

In many respects the process of recording and analysing the signals from the electrodes is similar to MEG; multiple signals are recorded, and the source models used for analysis are often identical [8] (source modelling is introduced in §2.3). However, there are some crucial differences. First, potentials must be measured relative to a reference. This could be obtained from another of the scalp electrodes, some combination of all of the scalp electrodes, or a reference electrode in a “neutral” location, such as the earlobe. The chosen reference clearly has a bearing on the outcome of the experiment. The second important difference is the need to take into account the conductive properties of head tissue. Analysis

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of EEG data relies on conduction models, which aim to predict how internal current sources give rise to surface potentials. This behaviour can be very complicated and difficult to predict, particularly since the skull is highly insulative, so the path of volume currents to the electrodes is convoluted and indirect. It also attenuates and filters the measured signals, particularly where the source is distant from the electrode. This impairs the signal-to-noise ratio. As a result, the accuracy with which sources can be localised is limited to a few centimetres [14].

The cost of the equipment required for EEG is particularly low in comparison to all of the other modalities reviewed herein. The temporal resolution is also excellent — better than 1 ms. The measured data are directly influenced by neural spiking, providing a clear insight into neuronal processes. It is particularly ideal for clinical applications where diagnoses can be made by identifying EEGs characteristic of certain pathologies. The ambiguity and limited accuracy of source localisation curbs the use of EEG for investigating functional mapping and connectivity. The procedure is more invasive than MEG due to the application of electrodes, which is time-consuming and intrusive.

2.2.4 Comparison

The sensitivity profile of magnetometer coils describes how greatly they respond to sources at different angles and distances from the sensor. It diminishes with distance, especially in the case of more conventional gradiometer-type sensors. Gradiometers are deliberately employed to measure the 1st-order field component so as to reject distant environmental noise. As a result, MEG is more suited to measuring superficial activity, since the signal-to-noise ratio of deeper sources is smaller. This contrasts with PET and MRI, which exhibit consistent performance over the entire experimental volume.

The data collected in an MEG experiment do not allow the signal from a specific region of interest to be isolated with ease. There are various ways of attempting this, which are discussed in the following section. However, external magnetic field measurements contain ambiguities that prevent the absolute separation of regional sources in the absence of prior

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information about the statistical relationships between them. Consequently, MEG cannot entirely replace invasive procedures such as electrocorticography (ECoG — the placement of microelectrodes onto the cortical surface during surgery). PET and fMRI have no such limitations, but the nature of the resulting information leaves ample territory for MEG investigations, which reveal far more about transient activity owing to their freedom from reliance on haemodynamic response.

The initial equipment costs for MEG are particularly high when compared with other modalities. The running costs are not insubstantial either, due to the consumption of liquid helium as a cryogen, which must be replenished on a regular basis [1].

With the exception of EEG, MEG is the only non-invasive functional neuroimaging modality capable of studying detailed electrical activity in the brain [7]. It is often stated that MEG has superior temporal resolution to PET and fMRI, but this does not fully describe the level of informational detail achieved. In particular, phase relationships between different sources can be studied due to the preservation of signal polarity. The transient characteristics of the signal are faithfully reproduced, something that no other non-invasive neuroimaging modality facilitates. It is as much the nature as the magnitude of the measured activity that is of interest in an MEG study; the accurate reproduction of transients in the MEG signal translates into an enhanced ability to draw comparisons between activations in different brain regions. The temporal resolution is only limited by the sample rate, which is typically similar to EEG. However, the low pass filtering effect of the brain tissue limits the potential for sample rate increases in EEG. It is theoretically possible to measure higher frequency activity using MEG, with the caveat that brain electrical activity has lower energy at such frequencies. The system signal-to-noise ratio then becomes the limiting factor.

Unlike the surface potentials measured by EEG, which are due to currents travelling via circuitous routes, the extra-cranial magnetic fields recorded in MEG pass directly through the tissue of the head. This is a unique advantage that improves the accuracy of source localisation. Under favourable conditions sources may be localised to within about 3 mm [14].

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2.3 MEG Analysis 34

MEG excels in terms of being both safe and non-invasive, being an entirely passive procedure, although a structural MRI scan is usually required to provide anatomical data with which the results of the MEG experiment are co-registered. That is, the alignment between the MEG sensor coils and the subject’s brain is determined for the purpose of modelling the source space, as discussed in the following section.

2.3 MEG Analysis

The fundamental purpose of functional neuroimaging is to identify which parts of a brain are active and at what times. In the case of MEG, this entails calculating neuronal activity from magnetic fields observed outside the head. In physical terms, the head can be viewed as a volume conductor, and neuronal activity as a time-varying current density. Helmholtz proved in 1853 that infinitely many internal current densities can produce a given magnetic field outside a volume conductor. Therefore, knowing the magnetic field is not sufficient to know for certain which current density produced it. Furthermore, MEG recordings do not describe a magnetic field completely. They merely sample the field in many places, leaving out further information that could distinguish between solutions. As a result of this uncertainty, there is no universally correct estimation procedure, and several are in contemporary use.

Simple methods take the form of source localisation, which attempts to identify the origin of an MEG signal to within a small brain region. Algorithms that perform source localisation generally start with a model for the magnetic field produced by neuronal currents, because this problem is more tractable. The forward model is a function that maps current densities, described by a set of parameters, to vectors of magnetic field measurements. The domain of the function is the source space, which includes all possible parameter combinations. The function’s range is the sensor space, a vector space with the same dimension as the number of sensors. Each orthogonal direction in the sensor space represents the output of a different sensor. To solve the original problem, an inverse solution is found by manipulating the parameters until the predicted and actual measurements are

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closest.

More advanced forms of source estimation allow for simultaneous activity taking place over prescribed surfaces or volumes, leading to tomographic maps of activity. The same underlying forward models are used nonetheless. Three methods are presented below that represent the state-of-the-art in MEG analysis. The model underpinning all of these methods has undergone various refinements since its introduction, which are also considered. Some comparisons will then be drawn between the methods.

2.3.1 Classic Model

The classic approach to MEG analysis is to model the neuronal activity as a single current dipole within a homogeneous, spherical conductor. A current dipole consists of a current source and sink of equal magnitude, separated by a small distance. As well as a location, the dipole has a magnitude and direction, known as the dipole moment, which is orientated from the sink to the source. The dipole moment is the product of the (scalar) current and the displacement vector between the source and sink. In the theoretical dipole, the displacement is infinitesimal and the current infinite, with the magnitude of the dipole moment being finite. Of course, a practical dipole has both finite displacement and current, but is closely approximated by a theoretical dipole when regarded from a sufficient distance [3, 8, 19]. The spherical conductor geometry facilitates analytical solution of the forward problem; i.e. what a dipole with given parameters would generate at the sensors. The following result is derived in Appendix A.

B(r) = µ0

4πF2(F Q × r0− Q × r0· r∇F ) (2.1)

where F = a(ra + r2− r0· r),

a = |r − r0|, r = |r|

B(r) is the magnetic field as a function of position, which is represented by the position vector r. Q is the dipole moment and r0 is the dipole location. The constant µ0 is the

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This is a 3-dimensional image of the magnetic field pattern from an experimental current dipole, recorded using an MEG instrument (4-D Neuroimaging Magnes 3600 WH). This is a whole-head device, so the sensor array is in the shape of a helmet. Blue/red represent fields orientated in/out of the head.

Figure 2.6: Experimental current dipole

by a current dipole. A notable property of this spherical conductor model is that a dipole orientated along any radius of the sphere does not produce a field at the sensors.

Interpretting MEG recordings with the help of this model calls for the dipole parameters to be estimated from a set of instantaneous field measurements as produced by the sensors. This inverse problem has no analytical solution, and must instead be solved numerically. A cost function is defined as the sum, over all sensors, of squared differences between each sensor measurement and the field predicted by the dipole model at the same location and orientation as that sensor. The cost function is then minimised for the dipole parameters using an iterative algorithm, eventually leading to an optimal least-squares fit. While this approach facilitates a solution, it unfortunately embodies some major assumptions that are difficult to justify.

Unlike a current dipole, practical neuronal currents are widely distributed around the brain, not infinitesimal in size. Although multiple trial averaging may be used to isolate a focal source, the spherical conductor remains a poor model for the brain volume. Sometimes a few dipoles are fitted to the data, and more sophisticated conduction models such as finite

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element or boundary element models can be used. However, the number of dipoles used dramatically affects their estimated locations; confidence in the results is not good without prior knowledge of the quantity of focal sources.

2.3.2 Minimum Norm Estimation

The dipole model introduced above can be greatly improved upon by modelling the current density as a dense grid of dipoles covering a putative source volume or surface. Instead of trying to estimate the location, direction and magnitude of a single dipole, a large complement of dipoles with fixed locations and directions is used. In this way, only the dipole amplitudes need to be estimated.

A predefined grid, chosen by the analyst, sets the dipole locations. Up to three dipoles are placed at each point. The use of three mutually orthogonal dipoles per grid point allows the direction of the estimated current density at a particular point to vary arbitrarily. The three dipoles each account for a different scalar component of the 3-dimensional current density vector. Alternatively, only one or two dipoles are placed at each point. In these cases, the dipole directions are based on assumptions about the signal source. For instance, the component of the current density radial to the spherical volume used in the forward model may be omitted due to its lack of influence on the measurements. Each location then has a perpendicular pair of dipoles in the tangential plane. The typical orientation of pyramidal cell dendrites within the cerebral cortex (detailed in §2.1.1) is commonly cited to justify the use of only one dipole, normal to cortex. Similarly, the grid locations themselves are distributed over relevant anatomical structures that are thought to contribute to the MEG signal.

The dipole amplitudes, here denoted j(t), are linearly related to the sensor signals by the lead field matrix, L. A row of L represents the sensitivity of an individual sensor to each dipole in the grid. The recorded signals, b(t), are a vector function of time, while e(t) represents additive measurement noise.

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L is calculated using the forward solution presented in the single dipole model. Each column, li, corresponds to a unique combination of dipole location (r0) and orientation

(Q, which is a moment with unit magnitude). The elements of liare calculated as follows:

li =         B(r₁) · o1 B(r₂) · o2 .. . B(r_m) · om         , (2.3)

where r1...m are the positions and o1...mthe orientations of the sensors. B(r) is the

three-dimensional vector-valued function given by Eq. 2.1. It is used here to predict the magnetic field at a sensor location r. A sensor’s output due to a source at r0 with direction and

strength Q is evaluated with the dot product B(r) · o. Subsequently Eq. 2.2 yields a set of linear equations to be solved for the dipole amplitudes.

Unfortunately, the number of amplitudes being estimated usually exceeds the number of measurement channels. The problem is underdetermined and has no unique solution (L is not invertible). The Moore-Penrose pseudoinverse can be used instead [20]:

ˆ_{j(t) = L}+_b(t) _(2.4)

where L+= LT LLT−1

This constrains the problem by also minimising the Euclidean norm of the solution, which is the square root of the sum of squared source amplitude estimates.

ˆ_j_{M N}_{(t) = argmin kˆ}_j(t)k _(2.5)

Owing to the choice of constraint, this method is known as Minimum Norm Estimation (MNE). The minimum norm estimate is optimal in the sense that it contains no energy that is not represented in the measurements [3, 20]. All other solutions consist of the minimum-norm solution plus a component that is invisible from the sensors’ point of view. Chapter 4 will provide a more comprehensive treatment of MNE.

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2.3.3 Beamforming

Like MNE, beamforming uses linear transformation to estimate sources of MEG signals. It also uses fixed dipoles — in this case a volumetric grid with three orthogonal components at each grid node. Beamforming differs from MNE in that spatial filters constructed from the signal covariance matrix are used to estimate the dipole amplitudes [21]. These filters can be derived in various ways, but a common type is the minimum-variance beamformer [22], described below.

First, a weighting matrix W is defined, which relates the dipole amplitudes to the source signals.

j(t) = WTb(t) (2.6)

The columns of the weight matrix, wi, represent individual spatial filters to be calculated

from the signal covariances, which are estimated over a chosen time window on the data. For discrete-time data, the estimated covariance matrix, S can be calculated as follows. The elements of b = (b1, b2, . . . , bm)Tare scalar functions of time. If the data window contains

measurements at time points t1...n, an m × n matrix of mean-centred measurements can be

defined: M =         b1(t1) − ¯b1 b1(t2) − ¯b1 · · · b1(tn) − ¯b1 b2(t1) − ¯b2 b2(t2) − ¯b2 · · · b2(tn) − ¯b2 .. . ... . .. ... bm(t1) − ¯bm bm(t2) − ¯bm · · · bm(tn) − ¯bm         (2.7) where ¯bi = 1 n n X j=1 bi(tj)

Now the data covariance matrix may be defined in terms of M

S = 1

nMM

T _(2.8)

An expression for the power, Pi, in each time course to be estimated can be written in terms

of the data covariance matrix and the weights.

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The weights are then determined by minimising Pi subject to the constraint that wiTli =

1, ∀ i, where li are the forward-predicted field patterns created by hypothetical sources at

each grid point. These vectors are also the columns of the lead field matrix defined in the previous section. The constraint sets the filter gain to unity at the region of interest while the gain is minimised elsewhere, creating a “virtual electrode” at the region of interest. The solution is given by wi = S−1_l i lT i S−1li (2.10)

Since the weights are dependent on the data, the method evidently relies on a statistical model to achieve signal separation.

2.3.4 Model Improvements

The three analysis methods described above rely on a forward model for predicting sensor signals from putative current densities. Conventionally, a conductive sphere is used to represent the head, primarily because this enables analytic solution of the forward problem (as in Appendix A). However, as the head is somewhat aspherical, the suitability of such a model is questionable, and it is easy to conceive of more realistic models that may require numerical solution but offer closer predictions of real-world MEG measurements. A better forward model corresponds to improved solutions to the inverse problem. To this endeavour, both boundary element (BEM) and finite element models (FEM) have been created which more accurately simulate the propagation of currents through the cephalic tissue. These inhomogeneous models are based on the anatomy of the subject, obtained from CT or MRI data. A realistic model might simulate different conductivities for the skull, brain, cerebrospinal fluid etc. These enhancements have been demonstrated to improve the accuracy of source localisation [23]. The improvement comes at the expense of considerably increased computation; solution of the most detailed FEM models requires the use of a supercomputer [24].

Analytic solutions are possible with certain conductor geometries because internal ohmic currents generate no magnetic field outside the conductor [25]. If the geometry is inaccurate,

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this result does not hold true. Experimental data suggests that volume currents do indeed produce a measureable field [23]. An alternative model uses a piecewise homogeneous conductor, partitioned into arbitrarily shaped volumes of different conductivity [24]. In this case, the effect of the volume currents is given by surface integrals over the interfaces between homogeneous regions. The boundary element method discretises these surfaces into numerous, usually triangular, elements. The integral over each discrete element can then be evaluated by a suitable approximation (linear, quadratic etc.). The accuracy is determined by the size of the surface elements and the chosen approximation.

Finite element models discretise volume rather than surfaces, permitting the simulation of inhomogeneous and anisotropic media [24]. Inhomogeneity could be modelled using BEM by the use of large numbers of homogeneous subvolumes with intricate surface detail, but the balance of computational efficiency is in favour of FEM in such instances. The results produced by FEM also tend to be more accurate than those of BEM. However, BEM is most efficient in simpler models [26].

Multiple Sphere Model

One means of improving the simpler spherical model is by substituting one of several overlapping spheres into the model for every sensor in the forward calculation [27]. The sphere that best approximates the head depends on the location and orientation of the sensor in question, so it is beneficial to adapt the parameters accordingly.

Two methods have been proposed for determining the sphere used for each sensor [27]. The first requires the sensitivity charactistics of the sensors (the lead fields) to be calculated on a predetermined grid using a boundary element method. For each sensor, the sphere centre is then adjusted so as to maximise the agreement between lead fields calculated using the multiple sphere model and the BEM model. The principal benefit in this case is the ability to estimate an ECD with an accuracy approaching that of the BEM model, without interpolating between the discrete points of the BEM lead fields. To achieve this using a boundary element model would require re-evaluation of the computationally intensive